The existence of nontrivial solutions to nonlinear elliptic equation of [formula omitted]– [formula omitted]-Laplacian type on [formula omitted

Abstract

In this paper, we study the following nonlinear elliptic equation of p – q - Laplacian type on R N : { − Δ p u + a ( x ) | u | p − 2 u − Δ q u + b ( x ) | u | q − 2 u = f ( x , u ) + g ( x ) , x ∈ R N u ∈ W ≡ W 1 , p ( R N ) ⋂ W 1 , q ( R N ) ( ⋆ ) where 1 < q ≤ p < N , and Δ s u = d i v ( | ∇ u | s − 2 ∇ u ) is the s - Laplacian of u . We prove that under suitable conditions on f ( x , t ) , if g ( x ) ≡ 0 and a ( x ) ≡ m > 0 , b ( x ) ≡ n > 0 for some constants m and n , then the problem ( ⋆ ) has at least one nontrivial weak solution (see Theorem 1.12), generalizing a similar result for p - Laplacian type equation in [J.F. Yang, X.P. Zhu, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded Domains(I)Positive mass case, Acta Math. Sci. 7 (1987) 341–359]. Also, we prove that under essentially the same assumptions on f ( x , t ) as that in Theorem 1.12, there exists a constant C > 0 , such that if ‖ g ‖ ∗ < C , then the problem ( ⋆ ) possesses at least two nontrivial weak solutions (see Theorem 1.15), generalizing a similar result in [D.M. Cao, G.B. Li, Huansong Zhou, The existence of two solutions to quasilinear elliptic equations on R N , Chinese J. Contemp. Math. 17 (3) (1996) 277–285] for p - Laplacian type equation. Since our assumptions on f ( x , t ) are weaker than that in the above-mentioned reference, Theorem 1.15 is better than the main result in the same even if p = q .